Hankel Determinant for a Subclass of Alpha Convex Functions

5/20/2018 Hankel Determinant for a Subclass of Alpha Convex Functions

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Bonfring International Journal of Data Mining, Vol. 4, No. 3, August 2014 1

Hankel Determinant for a Subclass of AlphaConvex Functions

Gagandeep SinghandGurcharanjit Singh

Abstract---In the present investigation, the upper bound of

second Hankel determinant 2342 aaa

for functions

belonging to the subclass of analytic functions is

studied. Results presented in this paper would extend the

corresponding results of various authors.

( )BAM ,;

Keywords--- Analytic Functions, Starlike Functions,

Convex Functions, Alpha Convex Functions, Subordination,

Schwarz Function, Second Hankel Determinant

I.

INTRODUCTION

LET Abe the class of analytic functions of the form

( )

=

+=2k

k

kzazzf (1.1)

In the unit disc { }: 1E z z= < .

By S we denote the class of functions ( ) Azf andunivalent inE.

Let U be the class of Schwarzian functions

( )

k

== 1kkzdzw

Which are analytic in the unit disc { }: 1E z z= zp .1)0(1 =pand Define the functionby

(zh

( ) ( ) ( ) ( )( )

( ) ( ) =

zfzzfz 32 ..1 321 ++++ zbzbzb (3.7)+=zfzf

zh 1

In view of the equations (3.4), (3.6) and (3.7), we have




( ) ( )

( )

++++

+++=

+

=

...2

...

1

13

32

21

33

221

1

1

zczczc

zczczc

zp

zpzh

+

++

+= ...

42

1

22

1

2

1 33

1213

22

121 z

ccccz

cczc

...3

8

313

2

2122

12

4

312132

1

2

4

212

2

21

22

1

2

111

+++++

+++=

zcBc

ccBc

cccB

zcBc

cB

zcB

Thus,

422;

2

212

21

21

211

1

cBcc

Bb

cBb +

==

And

(3.8).

82242

313

21

212

31

2131

3

cBcc

cBcccc

Bb +

+

+=

( )( )

Using (3.5) and (3.7) in (3.8), we obtain

( )

( ) ( )

( )

( ) ( ) ( ) ( ){ }[ ]

( )

( ) ( )( )

( ) ( ) ( )

( )

( ) 212

82772

516

21321

318

21113131

22

12

212

183

,12

12

+++++++

++

+++++

+++

++++++

+

++

++

+

++++

+++

=

+++++

++

++

=

+

=

c

B

A

AB

BA

ccB

A

c

BAa

cBA

cBAa

cBAa

( )( )

( )( )

( ) ( )( )( )

( )( ) ( )( )[ ]

( )( )( )( ) ( ) ( )

.

31

213

122

82772

1

2

1513

712142

83513

2

21761

24113

21322

1761

2118

31213

1484

,22

(3.9)

(3.9) yields,

( ) ( ) ( ){ }2244122213412

2342 cRNccMccLc

C

BAaaa ++

=

( ) ( ) ( ) ( ),31211192 24 +++=C

(3.10)

( ) ( ) ,2114 23 ++=LWhere

( ) ( ) ( )( )

( )( )( )[ ],2741312

28277214

252311221

316

+++

+++

++++++=

BAM

ISSN 2277 - 5048 | 2014 Bonfring




( )( )( ) ( ) ( ) ( )

( )( )( ) ( )( )( )

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )( )

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( )

( ) [ ]27413131

213

2313

5121312

214212

316

712

218212

1516

213151122

12

3163

316

2

21

231631

413

3313

221714

221

314

51312162

151216

2]

331371

22145131216[

++++

+++

++++++++

++++++

++++++++

++++

++++++

+++++++

+

++++++=

BA

AB

B

AN

And

( )( ) .1313 4 ++=R

Using Lemma 2.1 and Lemma 2.2 in (3.10), we get

( )( )

( )( )( )

( ) ( ) ( )

( )( ) ( )( )( )( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )( )( )

( )( ) ( )

( ) ( )

( ) ( )( ) ( ){ }

( )( )( ) ( )

( ) ( )

( )( ) ( )

( )

( ) ( )

( ) ( ) ( ) ( ){ } ( )

( ) ( ) ( )( )

.

21

2141

31

2218

214

22131

413

221

314

314

112

214

21

212

1516

221

318

314

16

2

31516

2312151

216

41

712

218

212

1516

21315112

21

2316

3316

2

21

231631

413

331371

2214

221314

2131516

212

1516

22342

zxcc

cx

c

cxcB

A

c

AB

B

C

BAaaa

+++

+++++

++

+++

++

+++

++

+

++++

+

+++

+++

+++

++++

+

++++

+++

+++

++++

+++

+

=

23

313712

214

2131516A

+++

+++

ISSN 2277 - 5048 | 2014 Bonfring




Assume that and , using triangular inequality andcc =1 [ 2,0c ] 1z , we have

( )( )

( )( )( ) ( ) ( )

( )

( )( ) ( )

( )( )( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )( )( )

( )( ) ( )

( ) ( )

( ) ( )( ) ( ){ }( )( ) ( ) ( )

( ) ( )( )( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( )( )

( ) ( ) ( )( )

.

21

24

31

2218

24

22

314

13

221

314

314

112

24

2

212

1516

2

21

3

18

314

162

31516

2312151

216

4

712

218

212

1516

21315112

21

2316

3316

2

21

231631

413

331371

2214

221

314

2131516

212

1516

23

313

712

2142131516

22342

+++

++

++++++

+++

++

+++++

+

++++

+

+++

+++

+++

++++

+

++++

+++

+++

++++

+++

+

+

+++++

cc

cc

cc

B

A

c

AB

B

A

C

BAaaa

( )( )

( ),

FC

=2

BAwhere 1= x and

( )

( )( )( )

( ) ( ) ( )

( )( ) ( )

( )( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )( )( )

( )( ) ( ) ( ) ( )

( ) ( )( ) ( ){ }( )( ) ( ) ( )( ) ( )

( )( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( )( )

( ) ( ) ( )( )2124312218

24

22

314

13

221

314

314

112

24

2

212

1516

221

318

314

162

31516

2

312151

2

16

4

712

218212

1516

21315112

21

2316

3316

2

21

231631

413

331371

2214

221

3142131516

212

1516

2331371

2214

2131516

+++

++

++++++

+++

++

+++++

+

++++

+

++++++

+++

++++

+

++++

+++

+++++++

+++

+

+++

+++

=

cc

cc

ccB

A

c

AB

B

A

F

Is an increasing function. Therefore ( ) ( ).1. FFMax =

Consequently

( )( )

( ),2

2342 cG

C

BAaaa

(3.11)

Where

( ) ( ).1FcG =

So

( ) ( ) ( ) ( ) ( ),31148 424 ++++= cQcPcG

( )Where P and ( )Q are defined in (3.2) and (3.3respectively.

Now

( ) ( ) ( )cQcP 23

+ c 4=G And

( ) ( ) ( ).22 QcPc + 12=G ( ) 0 cG = gives

( ) ( ) .024 =+ Qc 2 cP( )cG is negative at ( )

( ) .c =

2P

Qc=

ISSN 2277 - 5048 | 2014 Bonfring




So

( ) ( ).. cGcGMax =Hence from (3.11), we obtain (3.1).

The result is sharp for , andcc1 = 2212 = cc

( ).32113 = ccc

For and , Theorem 3.1 gives thefollowing:

1=A 1=B

( ) ( )Corollary 3.1.1: If Mzf , then

( )( )

( ) ( ),

31

1121

21

4

2

2342

+

+

+

Aaaa

( ) ( )( )

1 2

Where

( ) ( ) ( )( ) ( ) ( ) ( ).71221162741314 ++++++

0=

2273

23112

252512124 ++++++++=A

For , Theorem 3.1 gives the following result due to

Singh and Singh [16].

Corollary 3.1.2: If , then( ) ( )BASzf ,

( ).

4

2B

1=

2342

Aaaa

For , Theorem 3.1 gives the following result due toSingh and Singh [16].

Corollary 3.1.3: If , then( )BAKf ,

( ).

25

2

2

365122

5

576

2342

++

BAABBA

BABAABBAaaa

2

22

2162

++ BA

Putting 1,0 == A and in Theorem 3.1, weobtain the following result due to Janteng et al. [5].

1=B

( ) SzfCorollary 3.1.4: If , then

.12 a 342 aa

,1=Putting 1=A and in Theorem 3.1, weobtain the following result due to Janteng et al. [5].

1=B

fCorollary 3.1.5: If , then( ) Kz

.8

123

a

42

aa

REFERENCES

[1] D. G. Cantor, Power series with integral coefficients, Bull. Amer. Math.Soc., Vol 69, pp 362-366, 1963.

[2] P. Dienes,The Taylor Series, Dover, New York, 1957.[3] R. M. Goel and B. S. Mehrok,On the coefficients of a subclass of

starlike functions, Ind. J. Pure and Appl. Math., vol.12, no. 5, pp 634-647,1981.

[4] Aini Janteng, Suzeini Abdul Halim and Maslina Darus,Coefficient inequality for a function whose derivative has apositive real part, J. Ineq. Pure Appl. Math., vol.7, no.2, pp 1-5, 2006.

[5] Aini Janteng, Suzeini Abdul Halim and Maslina Darus,Hankdeterminant for starlike and convex functions, Int. J. Math. Anavol.1, no.13, pp 619-625,2007.

[6] Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Hankdeterminant for functions starlike and convex with respect symmetric points, J. Quality Measurement and Anal., vol.2, no.1,pp 343, 2006.

[7] R. J. Libera and E-J. Zlotkiewiez, Early coefficients of the inversof a regular convex function, Proc. Amer. Math. Soc., vol.8pp225-230, 1982.

[8] R. J. Libera and E-J. Zlotkiewiez, Coefficient bounds for thinverse of a function with derivative in P, Proc. Amer. MatSoc., vol.87, pp 251-257, 1983.

[9] T. H. MacGregor, Functions whose derivative has a positive repart, Trans. Amer. Math. Soc., vol.104, pp 532-537, 1962.

[10] B. S. Mehrok and Gagandeep Singh, Estimate of second Hankdeterminant for certain classes of analytic functions, Scientia Magn8(3), 85-94, 2012.

[11] P. T. Mocanu, Une proprit de convexit gnralise dala thorie de la reprsentation conforme, Mathematica, 11(34127-133, 1969.

[12] J. W. Noonan and D. K. Thomas, On the second Hankel determinaof a really mean p-valent functions, Trans. Amer. Math. Socvol. 223, no. 2, pp 337-346, 1976.

[13] Ch. Pommerenke, Univalent functions, Gttingen: Vandenhoeck anRuprecht., 1975.

[14] Gagandeep Singh, Hankel determinant for new subclasses

analytic functions with respect to symmetric points, Int. J. of ModerMath. Sci., vol.5, no.2, pp 67-76, 2013.

[15] Gagandeep Singh, Hankel determinant for a new subclass analytic functions, Scientia Magna, vol.8. no.4, pp 61-65, 2012.

[16] Gagandeep Singh and Gurcharanjit Singh, Second Hankdeterminant for subclasses of starlike and convex functions, TAppear.

ISSN 2277 - 5048 | 2014 Bonfring

Hankel Determinant for a Subclass of Alpha Convex Functions

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